Cutting a grid into boxes
Shikaku is a Japanese puzzle about dividing a grid into rectangles. Each rectangle holds one number, and that number is its area. Squares count too. The puzzle comes down to two questions: what shapes could this number take, and which number could reach this square?
Listing the shapes for each number
- Primes are strips. A 2, 3, 5 or 7 can only be a single row or a single column, because a prime has no other way to split into width times height. That cuts the options to two directions.
- Composite numbers have a few choices. A 4 can be 1 by 4, 4 by 1 or 2 by 2. A 6 can be 1 by 6, 2 by 3, 3 by 2 or 6 by 1. An 8 can be 1 by 8, 2 by 4, 4 by 2 or 8 by 1.
- The board trims the list. A 1 by 6 strip needs six squares in a line, so on the 5×5 board it is impossible. Any shape that would run off the edge or swallow a second number is out.
For each number, also consider every position, not just every shape. A 2 by 3 rectangle can sit in several places around its number. Count how many placements survive. Numbers with only one placement are your first moves.
A worked example on the 5×5 board
Say a 5 sits in the middle of the top row, and a 2 sits at the far right of that same row. A 5 is prime, so it is either a full row or a full column. A full top row would also swallow the 2, which is not allowed. If no other number sits in the 5's column, the 5 must run straight down it, all five squares. Draw it.
That rectangle now cuts the board in half. Every square left of it must belong to numbers on the left, and every square to the right to numbers on the right. The 2 at the top right now has fewer ways to fit, since it cannot reach across the 5. Often it has only one placement left, and the next rectangle falls into place.
Thinking from the empty squares
When the numbers stop giving answers, look at uncovered squares instead.
- Who can reach it? For an empty square, list every number whose rectangle could cover it. If only one can, that square belongs to it, which usually fixes its shape.
- Corners are tight. A board corner can only be covered by a rectangle that touches that corner. Few numbers can usually reach it.
- Leftover gaps must add up. When a region is fenced off by finished rectangles, the numbers inside it must add up to its number of squares. If they do not, one of your borders is wrong.
Common mistakes
- Trusting the solid fill too much. A rectangle fills in solid when its size matches its number, but that only means the size is right. It can still sit in the wrong place. Keep checking that it leaves room for its neighbours.
- Ignoring red rectangles. A rectangle turns red if it holds two numbers or none. Remove it with a tap rather than drawing around it.
- Committing to a big number early. Large numbers have the most shapes. Place the small and prime ones first, then let the big ones fill what is left.
- Drawing over finished work by accident. Drawing over existing rectangles replaces them. Undo straight away if a drag catches the wrong corner.
Building skill across the levels
Easy is 5×5 with rectangles of up to five squares, so the prime-number rule and corner checks often finish it. Medium is 7×7 with rectangles of up to six squares, and you will start needing the "who can reach this square" question. Hard is 9×9 with rectangles of up to eight squares, where most numbers have several possible shapes and the leftover-gap check becomes valuable.
Every board has exactly one solution. The hint first removes a rectangle that is not part of the solution, then draws a missing one and makes it glow. When it does, work out why that was the only fit before continuing. If you like carving a board into regions, Nurikabe at /games/nurikabe asks for a similar eye for space, with islands instead of rectangles.
